一类具有强奇性的矩阵型偏微分方程的正解的存在性

doi:10.7523/j.issn.2095-6134.2019.03.003

中国科学院大学学报 ›› 2019, Vol. 36 ›› Issue (3): 311-319.DOI: 10.7523/j.issn.2095-6134.2019.03.003

一类具有强奇性的矩阵型偏微分方程的正解的存在性

双震, 孙义静

中国科学院大学数学科学学院, 北京 100049

收稿日期:2018-01-22 修回日期:2018-04-13 发布日期:2019-05-15
通讯作者: 双震
基金资助:
国家自然科学基金（11571339，11771468）资助

Exsitence of positive solutions for matrix-type partial differential equations with strongly singular nonlinearities

SHUANG Zhen, SUN Yijing

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received:2018-01-22 Revised:2018-04-13 Published:2019-05-15

摘要/Abstract

摘要： 研究矩阵型强奇异偏微分方程

其中，Ω⊂Rⁿ是有界开集，M（x）是定义在Ω上的实对称矩阵，-p<-1，0 < q < 1，λ>0是参数，f（x）∈L¹（Ω），f（x）>0 a.e.in Ω。证明，如果存在u₀ ∈H₀¹（Ω）满足∫_Ωf（x）|u₀|^1-pdx <+∞，则对任意的λ>0上述方程都有正H₀¹-解，即慢速解。我们注意到，对于奇异方程，古典解即C²（Ω）∩C（Ω）解不一定是H₀¹（Ω）解。

关键词: H₀¹-解, 实对称矩阵, 强奇性

Abstract: We investigate the strongly singular partial differential equations of matrix-type,

where Ω is a bound and open set in Rⁿ, M(x) is a real symmetric matrix on Ω, -p<-1,0 < q < 1,λ>0 are parameters, f(x)∈L¹(Ω),f(x)>0 a.e. in Ω. We prove that the above-mentioned equation admits at least one positive H₀¹-solution when λ>0 if there exists u₀ ∈H₀¹(Ω) such that ∫_Ωf(x)|u₀|^1-pdx <+∞. It should be noted that a classical solution, namely, the C²(Ω)∩C(Ω)-solution, is not necessarily a H₀¹(Ω)-solution for singular equations.

Key words: H₀¹-solution, real symmetric matrix, strong singularity

中图分类号:

0175.29

双震, 孙义静. 一类具有强奇性的矩阵型偏微分方程的正解的存在性[J]. 中国科学院大学学报, 2019, 36(3): 311-319.

SHUANG Zhen, SUN Yijing. Exsitence of positive solutions for matrix-type partial differential equations with strongly singular nonlinearities[J]. , 2019, 36(3): 311-319.

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一类具有强奇性的矩阵型偏微分方程的正解的存在性

Exsitence of positive solutions for matrix-type partial differential equations with strongly singular nonlinearities

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